Pre-Quantum Electrodynamics

• Gr 6.1.3

Orbital motion of electrons: so fast that 'current' is for most purposes steady: \(I = e/T = \frac{ev}{2\pi R}\). Orbital dipole moment (\(I \pi R^2\)) is thus \[ {\bf m} = -\frac{evR}{2} \hat{\bf z} \label{Gr(6.4)} \] In practice: harder to tilt orbit than spin, so this represents a small paramagnetic contribution.

More significant: speeding up or slowing down of electron on its orbit. Here: handwaving calculation. Centripetal acceleration: \(v^2/R\) usually sustained by electrical forces alone, \[ \frac{1}{4\pi \varepsilon_0} \frac{e^2}{R^2} = m_e \frac{v^2}{R} \label{Gr(6.5)} \] In presence of magnetic field (say perpendicular to plane of orbit): \[ \frac{1}{4\pi \varepsilon_0} \frac{e^2}{R^2} + e \bar{v} B = m_e \frac{\bar{v}^2}{R} \label{Gr(6.6)} \] New speed: \(e \bar{v} B = \frac{m_e}{R} (\bar{v}^2 - v^2) = \frac{m_e}{R} (\bar{v} + v) (\bar{v} - v)\), or (assuming small change) \[ \Delta v = \frac{eRB}{2m_e} \label{Gr(6.7)} \] So electron speeds up when \({\bf B}\) is turned on. Change in magnetic dipole moment: \[ \Delta {\bf m} = -\frac{1}{2} e(\Delta v) R \hat{\bf z} = -\frac{e^2 R^2}{4m_e} {\bf B} \label{Gr(6.8)} \] so change in \({\bf m}\) is opposite to change in \({\bf B}\). Universal phenomenon, leading to diamagnetism. Usually much weaker than paramagnetism. Observed in substances having even number of electrons (where paramagnetism is usually absent).

Fun fact: a paramagnet attracted into the field, whereas a diamagnet is repelled away.